A hyperlink is a finite set of non-intersecting simple closed curves in $$\mathbb {R} \times \mathbb {R}^3$$ R × R 3 . Let S be an orientable surface in $$\mathbb {R}^3$$ R 3 . The dynamical variables in general relativity are the vierbein e and a $$\mathfrak {su}(2)\times \mathfrak {su}(2)$$ su ( 2 ) × su ( 2 ) -valued connection $$\omega $$ ω . Together with Minkowski metric, e will define a metric g on the manifold. Denote $$A_S(e)$$ A S ( e ) as the area of S, for a given choice of e. The Einstein–Hilbert action $$S(e,\omega )$$ S ( e , ω ) is defined on e and $$\omega $$ ω . We will quantize the area of the surface S by integrating $$A_S(e)$$ A S ( e ) against a holonomy operator of a hyperlink L, disjoint from S, and the exponential of the Einstein–Hilbert action, over the space of vierbeins e and $$\mathfrak {su}(2)\times \mathfrak {su}(2)$$ su ( 2 ) × su ( 2 ) -valued connections $$\omega $$ ω . Using our earlier work done on Chern–Simons path integrals in $$\mathbb {R}^3$$ R 3 , we will write this infinite dimensional path integral as the limit of a sequence of Chern–Simons integrals. Our main result shows that the area operator can be computed from a link-surface diagram between L and S. By assigning an irreducible representation of $$\mathfrak {su}(2)\times \mathfrak {su}(2)$$ su ( 2 ) × su ( 2 ) to each component of L, the area operator gives the total net momentum impact on the surface S.